Overdrawing Urns using Categories of Signed Probabilities
This work addresses a foundational problem in probability theory for researchers by providing a mathematical framework to handle overdraws, though it appears incremental as it extends existing distributions with negative probabilities.
The paper tackles the mathematical impossibility of overdrawing from an urn in probability theory by introducing a new extension of the hypergeometric distribution that allows overdraws using signed distributions with negative probabilities. The result is a conservative extension that enables draws of arbitrary sizes, leveraging dual basis functions of Bernstein polynomials and categorical probability frameworks.
A basic experiment in probability theory is drawing without replacement from an urn filled with multiple balls of different colours. Clearly, it is physically impossible to overdraw, that is, to draw more balls from the urn than it contains. This paper demonstrates that overdrawing does make sense mathematically, once we allow signed distributions with negative probabilities. A new (conservative) extension of the familiar hypergeometric ('draw-and-delete') distribution is introduced that allows draws of arbitrary sizes, including overdraws. The underlying theory makes use of the dual basis functions of the Bernstein polynomials, which play a prominent role in computer graphics. Negative probabilities are treated systematically in the framework of categorical probability and the central role of datastructures such as multisets and monads is emphasised.